The Reproducing Kernel Of Meyer Wavelet Transform Image Space

Wavelet analysis is a shining example of which converge the pure mathematics,applied mathematics and engineering together, at the same time, it provide apowerful tool for many fields. We all know that the continuous wavelet transform isthe basis of wavelet analysis. The image space of continuous wavelet transform is areproducing kernel Hilbert space. The elements of the reproducing kernel Hilbertspace are expressed by its reproducing kernel function, Because Reproducing KernelHilbert space is the basis of continuous wavelet transform, without reproducingkernel function, continuous wavelet transform can not be rebuilt. and not to exist thewavelet analysis. Therefore, to research reproducing kernel function is veryimportant basic work before describing the reproducing kernel space. In this paper,we will discussion the single week sine wavelet and Meyer wavelet.Meyer wavelet is a classic wavelet, it has many good properties. For examplederivation infinitely, smoothness, attenuates fast and its spectrum is finite, it iswidely used in signal detection, image processing, etc. In this paper, we get aspecific Meyer wavelet by selecting the appropriate smooth function. For the Meyerwavelet, Firstly, geting the Meyer wavelet is permissible wavelet. And to give theMeyer wavelet transform estimator and isometric identities too. Secondly, we get thereproducing kernel function expression of Meyer wavelet transform image space andproved that the reproducing kernel function is a continuous function. When the fixedscale factor, the reproducing kernel function of Meyer wavelet transform imagespace was further characterized. Finally, we given the sampling theorem of theMeyer wavelet transform image space.Weekly sine wavelet is a range of compactly supported wavelet, it has thenature of the infinitely differentiable, so weekly sine wave is a smooth wavelet.Because the sine wave is widely used in the field of signal processing. This paperwill give the reproducing kernel function expression of weekly sine wavelet.